Problem 137 - Calculus Volume 1

Question

For the following exercises, \(P\) is a point on the unit circle. a. Find the (exact) missing coordinate value of each point and b. find the values of the six trigonometric functions for the angle \(\theta\) with a terminal side that passes through point \(P .\) Rationalize denominators. $$P\left(x, \frac{\sqrt{7}}{3}\right), x<0$$

Step-by-Step Solution

Answer

The missing value is \(-\frac{\sqrt{2}}{3}\). Trig values: \(\sin = \frac{\sqrt{7}}{3}\), \(\cos = -\frac{\sqrt{2}}{3}\), \(\tan = -\frac{\sqrt{14}}{2}\), \(\sec = -\frac{3\sqrt{2}}{2}\), \(\csc = \frac{3\sqrt{7}}{7}\), \(\cot = -\frac{\sqrt{14}}{7}\).

Step 1: Understand the point on the unit circle

Given a point \(P(x, \frac{\sqrt{7}}{3})\) on the unit circle, recall that any point \((x, y)\) on the unit circle satisfies the equation \(x^2 + y^2 = 1\). We need to use this fact to find the value of \(x\).

Step 2: Solve for x

Substitute \(y = \frac{\sqrt{7}}{3}\) into the unit circle equation: \(x^2 + \left(\frac{\sqrt{7}}{3}\right)^2 = 1\). This simplifies to \(x^2 + \frac{7}{9} = 1\). Solve for \(x^2\) by subtracting \(\frac{7}{9}\) from both sides to get \(x^2 = 1 - \frac{7}{9} = \frac{2}{9}\). Thus, \(x = \pm\frac{\sqrt{2}}{3}\). Given that \(x < 0\), \(x = -\frac{\sqrt{2}}{3}\).

Step 3: Calculate sin(θ) and cos(θ)

For angle \(\theta\) with point \(P\left(-\frac{\sqrt{2}}{3}, \frac{\sqrt{7}}{3}\right)\) on the unit circle, calculate: \(\sin(\theta) = \frac{\sqrt{7}}{3}\) and \(\cos(\theta) = -\frac{\sqrt{2}}{3}\).

Step 4: Calculate tan(θ) and cot(θ)

Use the definitions \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{\sqrt{7}}{3}}{-\frac{\sqrt{2}}{3}} = -\frac{\sqrt{7}}{\sqrt{2}}\) and rationalize the denominator to get \(\tan(\theta) = -\frac{\sqrt{14}}{2}\). Then, \(\cot(\theta) = \frac{1}{\tan(\theta)} = -\frac{2}{\sqrt{14}} = -\frac{\sqrt{14}}{7}\).

Step 5: Calculate sec(θ) and csc(θ)

Use the definitions: \(\sec(\theta) = \frac{1}{\cos(\theta)} = -\frac{3}{\sqrt{2}} = -\frac{3\sqrt{2}}{2}\) and \(\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{3}{\sqrt{7}} = \frac{3\sqrt{7}}{7}\).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical tools that help us describe relationships within triangles and on the unit circle. There are six primary trigonometric functions: sine (\(\sin\theta\)), cosine (\(\cos\theta\)), tangent (\(\tan\theta\)), cosecant (\(\csc\theta\)), secant (\(\sec\theta\)), and cotangent (\(\cot\theta\)). These functions relate the angles of a triangle to the lengths of its sides.

• Sine measures the opposite side over the hypotenuse in a right triangle, or simply the y-coordinate on the unit circle.
• Cosine measures the adjacent side over the hypotenuse, or the x-coordinate on the unit circle.
• Tangent is the ratio of sine to cosine, \(\tan\theta = \frac{\sin\theta}{\cos\theta}\)
• Cosecant, \(\csc\theta = \frac{1}{\sin\theta}\), is the reciprocal of sine.
• Secant, \(\sec\theta = \frac{1}{\cos\theta}\), is the reciprocal of cosine.
• Cotangent, \(\cot\theta = \frac{1}{\tan\theta}\), is the reciprocal of tangent.

Understanding these functions helps in solving problems that involve angles and distances on the unit circle, providing a broad view of geometric and trigonometric concepts.

Rationalizing Denominators

Rationalizing the denominator is the process of eliminating any irrational numbers (like square roots) from the denominator of a fraction. This makes the expression easier to understand and work with in further calculations.

Consider the fraction \(\frac{\sqrt{a}}{b}\)

Multiply the fraction by the conjugate: \(\frac{\sqrt{a}}{b} \times \frac{\sqrt{b}}{\sqrt{b}}\)

The denominator is simplified to a rational number: \(b \times b = b^2\).

Adjust the expression accordingly: \(\frac{\sqrt{ab}}{b^2}\).

Rationalizing helps in making expressions cleaner and is particularly useful when further complex algebraic manipulations are needed.

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry, which states that for any angle \(\theta\), the square of sine plus the square of cosine equals one:\[\sin^2\theta + \cos^2\theta = 1\]This identity is derived from the Pythagorean theorem applied to the unit circle, where the radius is always 1. In terms of the unit circle,

• Sine represents the vertical leg.
• Cosine corresponds to the horizontal leg.

This identity can be used to find an unknown trigonometric function if the other is known. It's also essential in simplifying complex trigonometric expressions and proving other trigonometric identities. If you know that \(\sin\theta = \frac{\sqrt{7}}{3}\) and you use the identity, you can easily find \(\cos\theta\), as shown in the example above.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where geometric figures are represented by coordinates and equations. The unit circle, a perfect illustration of coordinate geometry, is a circle with a radius of one centered at the origin of a coordinate plane.

In coordinate geometry:

• Points are usually represented as ordered pairs \((x, y)\)
• Lines and curves are expressed in terms of equations.
• The unit circle equation is \(x^2 + y^2 = 1\)

In the context of this problem, achieving values that fit within the unit circle equation helps determine possible x-coordinates when a y-coordinate is given. This connects circle points to trigonometric concepts, facilitating the calculation of trigonometric functions for given angles.